I have $7$ unit vectors in $3$D space. Suppose we know the angles between some pairs of vectors. The graph below shows the known angles as edges. For example we know the angle between vectors '$3$' and '$2$', but not between '$3$' and '$5$'. Can I write all the unknown angles in terms of the known angles? If so how?
Also note that the angles made between the vectors $4,5,6,7$ (in total $^4C_2=6$ angles) can be taken to be equal to $\text{Cos}^{-1}(-1/3)=109.5$ degrees (the tetrahedral angle).
EDIT
This is part of an optimization involving $7$ vectors and my aim is to represent vectors in $3$D using just the angles formed between them. So if we have $7$ vectors we'll have $^7C_2$ such angles. But we dont need all of them to represent the vectors. Since its $3$D, each vector needs atleast $3$ known angles made with other vectors to be uniquely identified. If you look at the graph below, you'll see that each vertex have atleast $3$ edges connected to it. The vectors $4,5,6$ have four edges connected to it.
